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Generalize in Combinatorial Settings
Elise Lockwood, Zack Reed
To cite this version:
Elise Lockwood, Zack Reed. Leveraging Specific Contexts and Outcomes to Generalize in Combi- natorial Settings. INDRUM 2018, INDRUM Network, University of Agder, Apr 2018, Kristiansand, Norway. �hal-01849925�
Leveraging Specific Contexts and Outcomes to Generalize in Combinatorial Settings
Elise Lockwood and Zackery Reed
Oregon State University, Elise.Lockwood@oregonstate.edu
Generalization is a fundamental aspect of mathematics, and it is a practice with which undergraduate students should engage and gain fluency. It is important for students in combinatorial settings to be able to generalize, but combinatorics lends itself to engagement with specific examples, concrete outcomes, and particular contexts. In this paper, we seek to inform the nature of generalization in combinatorial settings by demonstrating ways in which students leverage specific, concrete settings to engage in generalizing activity in combinatorics. We provide two data examples that highlight ways in which concrete and specific ideas can be leveraged to help students develop generalizations in combinatorial settings.
Keywords: Combinatorics, Generalization, Examples, Discrete Mathematics.
INTRODUCTION AND MOTIVATION
Generalization is a foundational mathematical activity, a mathematical practice that both researchers and policy-makers value (Amit & Neria, 2008; Ellis, 2007). At the undergraduate level, given the nature of abstract, advanced mathematics, it is important to learn how to facilitate generalizing activity for students. We have recently conducted a study designed to investigate undergraduate students’
generalizing activity, and we explored the students’ generalizing activity in the context of combinatorial problems. In this study, we aimed to examine ways in which to foster students’ engagement in generalizing activity. In combinatorics, however, it is often important and even necessary to focus on specific contexts or to consider particular, concrete outcomes. Indeed, in our prior work (e.g., Lockwood 2013, 2014) and in this current study, we have found that concrete, specific instantiations of problems, outcomes, and examples are particularly important for students’ combinatorial thinking and activity. We believe that in the domain of combinatorics in particular, such specific instantiations are especially important for developing combinatorial thinking. Given that we want our mathematics students to be able to reason generally in combinatorial settings, we examine the interplay between the natural need for specific contexts and outcomes in combinatorics and the desire to have students engage in meaningful generalization. In this paper, we seek to answer the following research question: In what ways can specific examples, concrete outcomes, and particular contexts be leveraged to foster generalizing activity in a combinatorial setting?
LITERATURE REVIEW AND THEORETICAL PERSPECTIVE
A Piagetian perspective on generalization and generalizing activity. As a broad theoretical perspective, we adhere to a constructivist view of learning, asserting that students construct their knowledge of a given situation based on their mathematical experiences. We fundamentally view generalization as being related to Piaget’s notions of reflective abstraction, and we emphasize the importance of having students engage with and reflect upon their prior activity as they engage in generalization. Many researchers have studied generalization in a variety of contexts involving both school-aged children (Amit & Neria, 2008; Ellis, 2007; Rivera, 2010) and undergraduate students in a variety of areas (e.g., Dubinsky, 1991; Harel & Tall, 1991). This report contributes to the growing body of literature by examining the nature of generalization in an undergraduate combinatorial setting.
We follow Ellis (2011) and take generalization to mean engaging in “at least one of three actions: (a) identifying commonality across cases, (b) extending one’s reasoning beyond the range in which it originated, or (c) deriving broader results from particular cases” (p. 311). To describe students’ activity as they generalize, we adopt Ellis’ (2007) taxonomy of generalizing activity, Ellis describes three main categories of generalizing actions: relating, searching and extending. In this paper, we focus especially on relating, which occurs when “a student creates a relation or makes a connection between two (or more) situations, problems, ideas, or objects”
(p.235). In this paper, the term “generalization” need not involve a formal, final statement of a general rule or property, but rather it may refer to the results of a students’ generalizing activity, even if that activity is incomplete or not normatively correct.
Combinatorial thinking and activity.
Combinatorial enumeration problems, or “counting problems,” are easy to state, but they can be surprisingly challenging for students to solve. This is due in large part to the fact that counting problems are not reliably solved using prescribed, fool proof algorithms (e.g., Kapur, 1970). Solving counting problems thus provides opportunities for students at all levels to engage in rich mathematical thinking. There is ample evidence that students struggle with solving counting problems (e.g., Batanero, Navarro-Pelayo, & Godino, 1997). Although researchers have taken strides in identifying productive strategies and ways of thinking that might help address such difficulties, there remains much to learn about how we might effectively help students to count successfully.
In this paper, we examine the role of generalizing in students’ counting, and we explore how to frame generalizing activity in terms of Lockwood’s (2013) model of students’ combinatorial thinking. Lockwood (2013) suggested that there are three key components to students’ combinatorial thinking (Figure 1) and that solidifying the relationships between these components is an important aspect of successful
counting. Formulas/expressions refer to numbers and/or variables that represent the answer to a counting problem. Counting processes refer to the enumeration process in which a counter engages as they solve a problem. Sets of outcomes are the collection of (encoded) objects that are being counted.
Figure 1: Lockwood’s (2013) model of students’ combinatorial thinking
To exemplify the model, we discuss the Horse Race Problem, which is a problem that we discuss in the Results Section. The problem states: There are 10 horses in a race. In how many different ways can the horses finish in first, second, and third place? Note that one counting process to solve this problem is to consider options for which horse could be first, second, or third place. We can argue that there are 10 options for which horse is first, and for any choice of which horse is first there are 9 options for which horse is second, and then for any of those there are 8 choices for which horse is third. This counting process yields an expression of 10*9*8, which is 720. This process would organize the set of outcomes lexicographically, grouped according to which horse finished first, then second, then third.
Lockwood has emphasized the importance of sets of outcomes in a number of studies. In particular, she has advocated for a set-oriented perspective toward counting (Lockwood, 2014), in which the act of counting is viewed as inherently involving structuring and enumerating the set of outcomes. In addition, she has made a case for the value of listing outcomes, demonstrating that listing outcomes was positively correlated with solving problems correctly for novice undergraduate students (Lockwood & Gibson, 2016). In this way, Lockwood has emphasized the importance of considering concrete outcomes as students solve counting problems.
The point of the set-oriented perspective is that students should think carefully about what they should be solving in a given problem.
On one hand, then, this prior research suggests that it is useful for students to consider the concrete, specific outcomes that they should count. Further, when a student solves a counting problem such outcomes are necessarily tied to that problem and context. We want students to be able to think about what constitutes an outcome in a particular combinatorial situation. On the other hand, though, we want to foster generalization for students and to encourage them to engaging in generalizing activity, even in combinatorial situations. We want students to be able to develop and apply general formulas, or to be able to make general arguments about their counting processes. In this paper, we describe specific ways in which students use concrete settings to leverage general thinking and activity in the domain of combinatorics.
METHODS
We report on data from a study designed to study students’ generalizing activity in the context of combinatorics. We report on two data sources. First, we report on a design experiment (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003) with four undergraduate calculus students, and we focus on one particular student Carson (student names in this paper are pseudonyms). The students were chosen based on a selection interview; they had not taken a discrete mathematics course in the university and were novice counters who could explain their thinking. The students were interviewed together as a group of four during nine 90-minute sessions. The interviews were audio and video recorded. During this time, the students worked both individually and together on combinatorial activities, and the interviewer often asked probing questions or asked the students to explain their work. These tasks included solving basic counting problems, coming up with general formulas for counting problems, and solving problems related to combinatorial proof.
Second, we report on an individual interview with a calculus student, Tyler, who had similarly not taken a discrete mathematics course in the university. The interview was individual and 60 minutes long. We gave Tyler tasks involving determining the number of 3, 4, 5, and eventually n-length passwords using As and Bs, and then passwords consisting of the characters As, Bs, and the number 1. We had him write tables in which he recorded the number of passwords with a certain number of As, and ultimately the tasks could yield the binomial theorem (which we do not discuss in this paper). These tasks were broadly designed to target students’ generalizing activity in combinatorial tasks specifically, and we sought both to learn about students’ combinatorial reasoning and about their combinatorial generalization.
The design experiment sessions and the interviews were transcribed, and we created enhanced transcripts in which we inserted relevant images and descriptions of activity into the transcripts. For the purposes of this paper, we identified two cases of Carson and Tyler as students who leveraged particular problems and situations in order to engage in generalizing activity. We focused on these students’ data and identified relevant episodes that shed light on this phenomenon. We reviewed the transcripts and the videos and discussed these cases with the research team.
RESULTS AND DATA EXAMPLES
In our results, we seek to demonstrate instances in which students leveraged and use specific, concrete examples in combinatorics in order to engage in generalizing activity. These are meant to shed light on the interplay between particular situations and contexts that are important in combinatorial settings and the broader practice of generalization. We argue that specific examples, concrete outcomes, and particular contexts remain a fundamental aspect of combinatorial reasoning that can help to facilitate generalization. We offer two specific examples of how this phenomenon occurs, shedding light on the nature of generalization in combinatorial settings.
Students leverage activity on particular problems to generalize counting formulas and principles.
In this case, a student in the design experiment, Carson, repeatedly referred back to his prior work on a particular problem that stood out to him as being important. We view this as an example in which work on a particular problem can be leveraged to help students engage in generalizing activity. During initial problem solving in the first session, Carson had solved the Horse Race Problem, described previously. We will demonstrate that as he proceeded to solve other tasks and solve other problems, he repeatedly referred back to his prior activity on this problem, related it to other situations, and used it to generalize a counting formula. Carson solved the problem in a different way than we had described above, arriving at a correct expression of 10!/7!. Note, this is equivalent to the expression 10*9*8, but, as he explained below, Carson used a different counting process. He had a particular way of reasoning about this solution, leveraging the notion of division and equivalence to explain why the division by 7! makes sense combinatorially.
Carson: …So, there’s 10 factorial total outcomes, and then we know for any given first 3 there’s gonna be 7 factorial, because that’s saying we know the first 3 horses have finished – how can the last 7 horses finish, so that’s gonna be 7 factorial. But all we care about is how many given first 3s there are. So, if we divide the total number of outcomes by the number of potential of outcomes for the last 7 horses that will give us the potential number of outcomes for the first 3. If that makes sense?
Carson argued that for any particular arrangement of all ten horses, since all that matters is how the first three horses finish, we can divide by the number of identical arrangements of the last seven horses. This is a valuable way to think about these problems, and understanding and articulating this counting process seemed to be an important moment for Carson. As we proceeded to consider more problems, Carson repeatedly returned to this Horse Race Problem. We will demonstrate ways in which Carson has engaged in the generalizing activity of relating (Ellis, 2007) by using this problem as he approached additional tasks. In this way, this problem served as a generic example (Mason & Pimm, 1984), a way in which he could make general arguments and connect his reasoning to other problems. We now describe several of the ways that Carson leveraged this particular problem.
First, we see that Carson engaged in relating by connecting back to the Horse Race Problem as he solved other problems. For example, in solving a problem of arranging 4 of 7 books in a row on a shelf, Carson arrived at the correct answer of 7!/3!. The interviewer asked him how he was thinking about the problem, and his response below shows the connection he made to the Horse Race Problem.
Carson: Yes, kind of similar to the horse problem. You can say they’re all in a race, you wanna see how many ways the first 4 books could finish in the race.
We later had the four students categorize problems they had solved, and from that exercise we asked them to generalize formulas. One of these formulas was the number of ways of arranging some number of objects from a larger set of objects.
Carson had indicated that he saw several problems as being the same, and in the excerpt below, he explained why he viewed the problems as being the same. Again, he referred to the podium and the division that he had articulated on the Horse Race Problem as being a distinguishing feature of all of these problems.
Carson: So, essentially all of them are asking for a ranking of a given set of objects and asking how many arrangements there are for a given number of places, right? So, the cats are racing to get the collars you could say or the restaurants are racing to get the top five rankings in the town or the horses are racing in a race. Then each of the rankings or the collars are a ranking in the race. Yeah, then you can just divide by the duplicates for leftover ones, the ones that didn’t make the podium finish or whatever amount of finishes there are or whatever podium they’re asking for.
His reference to the horses and to the podium suggest to us that this continued to be a salient aspect of his reasoning. We interpret that Carson was engaging in the generalizing activity of relating (Ellis, 2007), and, in terms of Lockwood’s (2013) model, he related the counting process of arranging all of the objects and then dividing by the ways to arrange the leftover objects. He also seemed to emphasize the nature of the set of outcomes (arrangements). He recognized that counting process as similar among the problems he grouped together, and he related each of those other problems to the ranking and podium language he used in the Horse Race Problem.
Further, we also asked the students to come up with a general formula for the problems they had grouped together. They did this for several problems, but we highlight the formula for the permutation problems. In trying to articulate the kind of problem they were dealing with, again Carson referred to his activity on the Horse Race Problem.
Carson: Right. I mean thinking about the method for solving this, it’s the factorial from above, right? So, we have ten horses in a race. How many ways can the horses finish, but then how many of those have a unique podium, right?
So, how many times are the first, second, and third place different?
The students then had a conversation about what the formula would be. They came up with the formula a!/(a-b)! for arranging b objects from a set of a distinct objects, which another student, Josh, articulated:
Josh: No, I think that it would be something like if you have a objects, you would have a factorial – that’s the total number of things that you can select – over a minus b factorial, where b is the number of slots that you have.
After they agreed upon this formula, Carson explained how he was thinking about this general formula they had come up with. The excerpt below shows that he referred back to the imagery of the podium, using that context to make a general argument.
Carson: … A is your total number of arrangements for the entire thing and then you want to divide by the number of ways that the places you’re not selecting can be arranged, right? So, if you’re selecting first, second, and third, then you have fourth through tenth and those can be arranged in ten minus three factorial ways, right? So, we can just divide by that number of arrangements [begins motioning slots with hands] for the back end to get just one for the front end because that’s what we’re asking for is how many ways can that podium be arranged?
We contend that in relating back to the Horse Race Problem, Carson was relating back to different components of Lockwood’s (2013) model, including formulas, sets of outcomes, and counting process. This exchange suggests that Carson had a well- developed understanding of the specific problem in terms of the components of the model, and he related different aspects of the problem in different situations. From our Piagetian perspective we view Carson and the students as constructing a formula that makes sense to them, and Carson reflected upon his prior activity in order to develop a statement of a more general formula.
The Horse Race Problem came up in additional settings for Carson, including during reasoning about combinatorial proof in a later session. Ultimately, Carson acknowledged how important this problem was for him. In the final interview, when we were reflecting on the entire design experiment, Carson shared that he continued to think about subsequent problems in terms of the Horse Race Problem. We interpret that his language below means that he felt that he conceptually understood the ideas in the Horse Race Problem, perhaps in a deep way that he felt confident about.
Carson: For whatever reason, the horse race problem is the one that’s in my head forever. And it must have just been where it clicked in the interview because that’s kind of what I refer to. If somebody says how many ways can a horse finish in the podium, how many ways can the podium be organized, things like that. And that’s kind of where I keep going back to. And I don’t know why that is.
This case serves as an example of a student leveraging activity on a particular problem for a number of other activities, particularly generalizing activities of relating (Ellis, 2007). There have other examples of students drawing on prototypical problem types in combinatorics. Maher and colleagues have talked about students referring to Pizza Problems or Block Towers problem, demonstrating how students think about and use powerful particular problems in other (e.g., Maher, Powell, &
Uptegrove, 2011). We build on such work by explicitly drawing attention to the
generalizing activity that a specific problem fostered for students, highlighting the affordances that can stem from a student deeply understanding and justifying his or her activity on a particular counting problem.
Students compare and contrast specific examples to identify general structure.
We briefly describe an additional example in which a calculus student Tyler was relating two different situations while working on the Passwords Activity. In the interview Tyler was counting two kinds of passwords – those involving either upper case As and/or Bs, and those involving As, Bs, and the number 1 (with repetition allowed). Tyler had initially engaged in systematic listing activity to count the number of possible 4-character AB passwords. In particular, he created the list of 4- character AB passwords with exactly two As (Figure 2a), and the table of passwords according to number of As (Figure 2b).
Figure 2a, b: Tyler’s arrangements of 4-character AB passwords with exactly 2 As and his complete 4-character AB table
Later in the interview, Tyler was in a situation of counting 4-character passwords consisting of uppercase As, Bs, and 1s. We had asked him to create a table based on the number of 1s in the password. To complete this table, one can first consider placing the 1(s) and then filling the remaining positions with either A or B. Notably, placing As and Bs then reduced to the prior problem Tyler had solved, namely counting 4-character AB passwords. Tyler realized that there were the same number of arrangements of two types of characters, and he made a general statement about this case, recognizing that he will always have six ways of arranging two kinds of characters. Tyler was able to speak generally about counting arrangements of two kinds of characters. That is, he recognized that the tables gave him totals for the number of ways of arranging two characters, not just that they had to be As and Bs or 1s and xs. In the excerpt below he had been working on an extension problem, and he speaks about two different “things” that are changing, suggesting he had extrapolated a notion of arranging two characters independently of what the characters are specifically.
Tyler: Um well these are all the number of combinations I can do, um, with 2 different, 2 things that are changing, and this number of letters.
Here, we conjecture that reasoning about the particular situations and engaging with the actual outcomes allowed Tyler to make an important connection between arranging As and Bs and 1s and xs. The similar nature of the activity when listing in both cases allowed him to draw attention to the similar counting process in which he was engaging and the fact that the outcomes he was generating were fundamentally similar – arrangements of two kinds of characters. Ultimately this allowed him to make and use a useful generalization, and he understood the values in the rows in the AB tables as representing the number of arrangements of two kinds of characters.
We infer that Tyler engaged in relating (Ellis, 2007), and that comparing both situations allowed him to draw out some general commonalities between the two specific settings. Here, we argue that reasoning carefully about the concrete examples and actually engaging in concrete listing activity may have helped to solidify a broader combinatorial process.
CONCLUSION AND DISCUSSION
Prior research (e.g., Lockwood, 2013; 2014; Lockwood & Gibson, 2016) has emphasized the importance of having students focus on sets of outcomes as they solve counting problems. Often this focus on outcomes necessarily means that students reason about very specific contexts and concrete objects, and we view this as a fundamental aspect of counting and combinatorial activity. However, we also acknowledge that part of mathematical engagement involves looking beyond particular situations and contexts, and we have tried to demonstrate certain ways in which the particular contexts and concrete outcomes can be leveraged to facilitate meaningful generalizing activity for students.
Specifically, we offer two qualitatively different examples in which students leveraged the structure of specific combinatorial contexts to establish more general relationships. In our first example, Carson used his activity on and solution of the Horse Race Problem as a template for a specific combinatorial process, which he then used in similar contexts. Carson’s generalizing activity was manifest through using this template as a means to relate combinatorial processes that he viewed as similar in some way and to connect the structure of the Horse Race problem to other cases. This specific generalizing activity demonstrates a powerful manifestation of relating, wherein Carson leveraged the structure of a known specific example as a solution for novel and abstract counting processes.
In our second example, Tyler used a connection between two situations to generalize a concrete arrangement structure. Tyler had initially recognized that his tables partitioning the 3-length passwords according to the number of As represented instead a partition of ways to arrange pairs of objects. Tyler then leveraged the specific process with which he was familiar by arranging 1s and xs, thus implementing the same specific counting process in a particular generalized setting.
Both of these examples involve the activity of using concrete situations to form a general relationship. These cases help to inform the nature of generalization in combinatorial contexts, offering examples of specific ways in which concrete outcomes and situations can be leveraged for use in more general settings.
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